# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/mathematics/vector-operations/vector-coordinate-system Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/vector-operations/vector-coordinate-system/en.mdx Learn Cartesian coordinates in 2D and 3D space. Learn position vectors, displacement vectors, notation methods, and real-world GPS applications. --- ## Cartesian Coordinate System To specify the position of an object, such as an aircraft in flight, we need a **frame of reference** or **coordinate system**. The most commonly used system is the **Cartesian Coordinate System**, developed by René Descartes. The Cartesian Coordinate System uses perpendicular axes to determine the location of a point. - In **two dimensions**, we use the $$x$$ axis (horizontal) and $$y$$ axis (vertical). A point's position is expressed as an ordered pair $$(x, y)$$, for example, point $$P(3, 4)$$. This means that point $$P$$ is located $$3 \text{ units}$$ to the right (positive $$x$$ axis) and $$4 \text{ units}$$ upward (positive $$y$$ axis) from the origin $$O(0, 0)$$. Position Vector $$P(3, 4)$$ } description={ <> Origin $$O(0, 0)$$ to point{" "} $$P(3, 4)$$. } vectors={[ { id: "OP", name: "OP", points: [ { x: 0, y: 0 }, { x: 3, y: 4 }, ], }, ]} /> - In **three dimensions**, we add a $$z$$ axis that is perpendicular to the $$xy$$ plane. A point's position is expressed as $$(x, y, z)$$, used for determining the position of objects like aircraft in space. We can think of $$x$$ as length, $$y$$ as width, and $$z$$ as height. Position Vector $$P(3, 4, 5)$$ } description={ <> Origin $$O(0, 0, 0)$$ to point{" "} $$P(3, 4, 5)$$. } vectors={[ { from: [0, 0, 0], to: [3, 4, 5], color: getColor("PURPLE"), label: "P(3, 4, 5)", }, ]} /> Visible text: - In **two dimensions**, we use the axis (horizontal) and axis (vertical). A point's position is expressed as an ordered pair , for example, point . This means that point is located to the right (positive axis) and upward (positive axis) from the origin . Position Vector } description={ <> Origin to point{" "} . } vectors={[ { id: "OP", name: "OP", points: [ { x: 0, y: 0 }, { x: 3, y: 4 }, ], }, ]} /> - In **three dimensions**, we add a axis that is perpendicular to the plane. A point's position is expressed as , used for determining the position of objects like aircraft in space. We can think of as length, as width, and as height. Position Vector } description={ <> Origin to point{" "} . } vectors={[ { from: [0, 0, 0], to: [3, 4, 5], color: getColor("PURPLE"), label: "P(3, 4, 5)", }, ]} /> ## Difference Between Vectors and Coordinates Unlike coordinates which represent the _position_ of a point, a **vector** is a quantity that has both **magnitude (length)** and **direction**. Imagine a fly traveling from point $$P(3, 4)$$ to point $$Q(-2, -8)$$. The fly's displacement can be represented as a vector. Visible text: Unlike coordinates which represent the _position_ of a point, a **vector** is a quantity that has both **magnitude (length)** and **direction**. Imagine a fly traveling from point to point . The fly's displacement can be represented as a vector. - **Position Vector:** A vector starting from the origin $$O(0, 0)$$ to a point $$P(x, y)$$. The position vector of $$P$$ is written as $$\overrightarrow{OP}$$. Example: $$\overrightarrow{OP} = (3, 4)$$. - **Displacement Vector (or Free Vector):** A vector connecting any two points, for example, from point $$P(x_P, y_P)$$ to point $$Q(x_Q, y_Q)$$. This vector is written as $$\overrightarrow{PQ}$$. It doesn't have to start from the origin. Example: displacement vectors from $$P(3, 4)$$ to $$Q(-2, -8)$$ or from $$Q(-2, -8)$$ to $$R(1, 3)$$. Displacement vectors from $$P$$ to{" "} $$Q$$ or from $$Q$$ to{" "} $$R$$ } vectors={[ { id: "QR", name: "QR", points: [ { x: -2, y: -8 }, { x: 1, y: 3 }, ], }, { id: "PQ", name: "PQ", points: [ { x: 3, y: 4 }, { x: -2, y: -8 }, ], direction: "backward", }, ]} /> Visible text: - **Position Vector:** A vector starting from the origin to a point . The position vector of is written as . Example: . - **Displacement Vector (or Free Vector):** A vector connecting any two points, for example, from point to point . This vector is written as . It doesn't have to start from the origin. Example: displacement vectors from to or from to . Displacement vectors from to{" "} or from to{" "} } vectors={[ { id: "QR", name: "QR", points: [ { x: -2, y: -8 }, { x: 1, y: 3 }, ], }, { id: "PQ", name: "PQ", points: [ { x: 3, y: 4 }, { x: -2, y: -8 }, ], direction: "backward", }, ]} /> ## Vector Notation Vectors can be written in several ways: - **Row Vector:** Using regular parentheses. Example: $$\overrightarrow{OP} = (3, 4)$$. - **Column Vector:** Using square brackets or large parentheses. Example: $$\overrightarrow{OP} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$$. - **Unit Vector Notation for Three Dimensions:** Using a combination of unit vectors $$\mathbf{i}$$, $$\mathbf{j}$$, and $$\mathbf{k}$$ which are aligned with the $$x$$, $$y$$, and $$z$$ axes. Example: a vector $$\overrightarrow{AB}$$ can be written as $$\mathbf{i} - 3\mathbf{j} - 4\mathbf{k}$$, meaning vector $$\overrightarrow{AB}$$ has a component of $$1$$ in the $$x$$ direction, $$-3$$ in the $$y$$ direction, and $$-4$$ in the $$z$$ direction. Visible text: - **Row Vector:** Using regular parentheses. Example: . - **Column Vector:** Using square brackets or large parentheses. Example: . - **Unit Vector Notation for Three Dimensions:** Using a combination of unit vectors , , and which are aligned with the , , and axes. Example: a vector can be written as , meaning vector has a component of in the direction, in the direction, and in the direction. ## Why Are Coordinate Systems Important? Imagine a map application on your phone. This application relies on coordinate systems to: 1. Determine your current location. 2. Find your destination. 3. Calculate the shortest or fastest route. 4. Provide direction instructions (turn left, turn right, go straight). Visible text: 1. Determine your current location. 2. Find your destination. 3. Calculate the shortest or fastest route. 4. Provide direction instructions (turn left, turn right, go straight). Without coordinate systems, it would be impossible for applications to accurately understand position and direction. Every movement, such as an aircraft changing altitude or direction, needs to be reported and processed using coordinate systems and vector concepts.